1. Direct Method of Interpolation
Civil Engineering Majors
Authors: Autar Kaw, Jai Paul
Transforming Numerical Methods Education for STEM Undergraduates

2. Direct Method of Interpolation http://numericalmethods.eng.usf.edu

3. What is Interpolation ?
Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.
Figure 1 Interpolation of discrete.

4. Interpolants Evaluate Differentiate, and Integrate
Polynomials are the most common choice of interpolants because they are easy to:
Evaluate
Differentiate, and
Integrate

5. Direct Method Given ‘n+1’ data points (x0,y0), (x1,y1),………….. (xn,yn),
pass a polynomial of order ‘n’ through the data as given
below:
where a0, a1,………………. an are real constants.
Set up ‘n+1’ equations to find ‘n+1’ constants.
To find the value ‘y’ at a given value of ‘x’, simply substitute the value of ‘x’ in the above polynomial.

6. Example
To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the direct method for linear interpolation.
Temperature vs. depth of a lake

7. Linear Interpolation Solving the above two equations gives, Hence

8. Example
To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the direct method for quadratic interpolation.
Temperature vs. depth of a lake

9. Quadratic Interpolation
Solving the above three equations gives

10. Quadratic Interpolation (contd)
The absolute relative approximate error obtained between the results from the first and second order polynomial is

11. Example
To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the direct method for cubic interpolation.
Temperature vs. depth of a lake

12. Cubic Interpolation

13. Cubic Interpolation (contd)

14. Comparison Table

15. Thermocline
What is the value of depth at which the thermocline exists?
To find the position of the thermocline we must find the points
of inflection of the third order polynomial, given by: .

16. Additional Resources
For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

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